2
2
CO
2
: Real-time time-dependent
electronic-structure approaches
Cite as: J. Chem. Phys. 154, 014101 (2021); https://doi.org/10.1063/5.0033072
Submitted: 13 October 2020 . Accepted: 11 December 2020 . Published Online: 04 January 2021
Carlo Federico Pauletti, Emanuele Coccia, and Eleonora Luppi
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Fully variational incremental CASSCF
Role of exchange and correlation
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of H
2
, N
2
, and CO
2
: Real-time time-dependent
electronic-structure approaches
Cite as: J. Chem. Phys. 154, 014101 (2021);doi: 10.1063/5.0033072Submitted: 13 October 2020 • Accepted: 11 December 2020 • Published Online: 4 January 2021
Carlo Federico Pauletti,1,2 Emanuele Coccia,1 and Eleonora Luppi2,a)
AFFILIATIONS
1Dipartimento di Scienze Chimiche e Farmaceutiche, Via Giorgieri 1, Trieste Italy
2Laboratoire de Chimie Théorique, Sorbonne Université and CNRS, F-75005 Paris, France
a)Author to whom correspondence should be addressed:eleonora.luppi@upmc.fr
ABSTRACT
This study arises from the attempt to answer the following question: how different descriptions of electronic exchange and correlation affect the high-harmonic generation (HHG) spectroscopy of H2, N2, and CO2molecules? We compare HHG spectra for H2, N2, and CO2with differentab initio electronic structure methods: real-time time-dependent configuration interaction and real-time time-dependent density functional theory (RT-TDDFT) using truncated basis sets composed of correlated wave functions expanded on Gaussian basis sets. In the framework of RT-TDDFT, we employ Perdew-Burke-Ernzerhof (PBE) and long-range corrected Perdew-Burke-Ernzerhof (LC-ωPBE) func-tionals. We study HHG spectroscopy by disentangling the effect of electronic exchange and correlation. We first analyze the electronic exchange alone, and in the case of RT-TDDFT with LC-ωPBE, we use ω = 0.3 and ω = 0.4 to tune the percentage of long-range Hartree– Fock exchange and short-range exchange PBE. Then, we added the correlation as described by the PBE functional. All the methods give very similar HHG spectra, and they seem not to be particularly sensitive to the different description of exchange and correlation or to the correct asymptotic behavior of the Coulomb potential. Despite this general trend, some differences are found in the region connecting the cutoff and the background. Here, the harmonics can be resolved with different accuracy depending on the theoretical schemes used. We believe that the investigation of the molecular continuum and its coupling with strong fields merits further theoretical investigations in the near future.
Published under license by AIP Publishing.https://doi.org/10.1063/5.0033072., s
I. INTRODUCTION
The optical response of a molecular system in intense ultra-short laser fields is a subject of increasing interest since the advent of attosecond (10−18 s) laser pulse generation, characteri-zation, and application.1–4In fact, the recent impressive advances in laser technology are continuously triggering the introduction of new time-resolved spectroscopies, which offer the opportu-nity to investigate electron dynamics with unprecedented time resolution.5–11
Attosecond pulses may be obtained via the nonlinear optical process high-harmonic generation (HHG), which can be understood semi-classically as a sequence of three steps (three-step model, 3SM): (1) electron ionization in a strong infrared (IR) field, (2) electron acceleration due to the laser field, and (3) electron recombination with the parent ion. During the recombination, coherent extreme ultraviolet (XUV) and soft x-ray radiations with a sub-femtosecond temporal resolution, i.e., HHG, are emitted.12,13
the system, a non-stationary electronic wavepacket, consisting of a coherent superposition of excited states, is generated. The time-evolution of the wavepacket involves changing interference and coupling between the different excited states. Moreover, the wavepacket dynamics is determined by parameters of the laser such as intensity, duration, polarization, and phase of carrier frequency.
The proper treatment of the time-dependent electronic wavepacket, and therefore of the many-electron dynamics, under the influence of the laser field is obtained by propagating the dependent Schrödinger equation (TDSE). Real-time time-dependent electronic-structure approaches can be conceptually separated in two classes: (1) real-time time-dependent wave func-tion (RT-TDWF) based methods18–35 and (2) real-time time-dependent density functional theory (RT-TDDFT).36–46 In RT-TDWF, the many-electron dynamics is described by a correlated time-dependent wave function, while in RT-TDDFT, the key quan-tity is the time-dependent density.
Another essential theoretical/computational aspect, common to RT-TDWF and RT-TDDFT, is the strategy used to solve the TDSE. In fact, time propagation can be directly applied to the molecular orbitals (and the amplitudes)26,31–33 or to a truncated basis composed of the ground- and excited-state correlated wave functions of the field-free electronic Hamiltonian.18,19,21,47–49In the last mentioned approach, for RT-TDDFT, the excited states are derived in the linear-response Kohn–Sham (KS) framework.50 In practice, the wave functions are never explicit, but only the TDDFT excitation energies and transition dipole moments are used in the propagation.18
In the framework of RT-TDWF, most of the theoretical approaches developed are given by the time-dependent extension of the well-established methods: configuration interaction (CI), cou-pled cluster (CC), and multiconfigurational self-consistent field (MCSCF).26,27,31,35,51–53 Recently, RT-TDWF methods have been applied to investigate HHG in atoms and molecules. The importance of electron correlation in HHG for He, Be, and Ne atoms was inves-tigated by Satoet al.27using the real-time time-dependent complete-active-space self-consistent-field (RT-TD-CASSCF) method and for Ne and Ar atoms by Pathak et al.53 with the time-dependent coupled-electron pair approximation with optimized orbitals (TD-OCEPA0) method. Luppi and Head-Gordon18 also investigated the role of electron correlation in H2 and N2 by using real-time real-time-dependent configuration interaction with single (RT-TD-CIS) and perturbative-double [RT-TD-CIS(D)] excitations and real-time time-dependent equation-of-motion coupled-cluster sin-gles and doubles (RT-TD-EOM-CCSD) methods. RT-TD-CIS has been used by some of us to describe the Cooper minimum in the HHG spectrum of an Ar atom at different laser intensities.54 The role of electron correlation in HHG of CO2has been investigated by means of the real-time time-dependent propagation of algebraic-diagrammatic construction (ADC) states.24 The role of HHG as a probe for isomers of polyatomic organic molecules was investi-gated by Bedurkeet al.29using RT-TD-CIS and by Wong et al.28 using a multielectron wave function coupled to the Cartesian grid approach.
In the framework of RT-TDDFT, the real interacting many-electron system is usually described by a non-interacting KS system able to reproduce the same density. In the case of a
direct propagation, the TDDFT solution of the TDSE for the real system is replaced by the solution of a time-dependent equation, which propagates the time-dependent KS orbitals.55,56 The many-electron effects are encoded in the time-dependent exchange–correlation potential vxc, which is a functional of the density, and also, in principle, depends on all previous times. However, in most of the cases, the adiabatic approximation is used, i.e., vxc is evaluated at the instantaneous time-dependent density.57
To accurately describe the strong-field electron dynamics in RT-TDDFT, it is necessary to correctly reproduce the long-range behavior of vxc. This permits to reproduce the ionization thresh-old energy, giving the onset of the continuum spectrum. Differ-ent strategies have been introduced such as self-interaction cor-rections (SIC),58,59range-separated functionals,60–63and long-range corrected potentials.64,65
Instead, the RT-TDDFT approaches that propagate a truncated eigenstate basis, constructed from linear-response TDDFT, describe the many-electron effects through the exchange–correlation kernel fxc.fxc is the functional derivative of vxc with respect to the den-sity and also needs to be approximated.fxcis nonlocal in time and space, but the most common approximations are adiabatic, and only the nonlocality in space is taken into account. In particular, range-separated approaches are among the most successful schemes to model the space dependence. Usually, in these approaches, the exchange part offxc, i.e., fx, is decomposed in a long-range (lr) Hartree–Fock (HF) and a short-range (sr) DFT component: fx = fx,HFlr +fxsr. The long-range nonlocal exchange kernel permits to better describe Rydberg and ionization potentials, but the lack of frequency-dependence prevents the treatment of doubly excited states.57
Recently, RT-TDDFT has been applied to investigate HHG molecules. Monfaredet al.66studied the effects of inner orbitals in the HHG spectra for N2O and, in particular, the role of valence elec-trons as a way to extend the harmonic plateau by using TDDFT with SIC correction to vxcin the local-density approximation (LDA). The role of inner-valence molecular orbitals was also studied by Chu and Groenenboom67for the HHG spectra of N2. In this case, the LBα
exchange–correlation potential was used for a correct representation of the continuum states. Gormanet al.68investigated the structural properties of CO2, N2O, and OCS molecules in combination with HHG spectroscopy using TDDFT with SIC corrections to the LDA vxc. Luppi and Head-Gordon18also investigated the role of electron correlation in H2and N2by using RT-TDDFT.
In this work, we have investigated the role of exchange and cor-relation in the HHG spectra of H2, N2, and CO2molecules (at fixed internuclear separation) with RT-TD-CIS and RT-TDDFT using a truncated basis composed of the ground- and excited-state corre-lated wave functions of the field-free electronic Hamiltonian. This basis is represented using Gaussian functions adapted for strong-field processes.69At the RT-TDDFT level, we have studied the effect of the exchange and correlation on the HHG spectra by means of the Perdew-Burke-Ernzerhof (PBE)59 and range corrected long-range corrected Perdew-Burke-Ernzerhof (LC-ωPBE) functionals, i.e., by means of RT-TD-PBE and RT-TD-LC-ωPBE.60,61The
This article is organized as follows: in Sec. II, we introduce the formalism of RT-TD-CIS and RT-TDDFT. The computational method is described in Sec. III, and the results are discussed in Sec.IV. Finally, in Sec.V, conclusions and perspectives are given. Unless otherwise indicated, atomic units are used throughout this paper.
II. THEORY
The time-dependent Schrödinger equation for a molecular system perturbed by an external time-dependent electric field is given by
i∂∣Ψ(t)⟩
∂t = ( ˆH0+ ˆV(t))∣Ψ(t)⟩, (1) where ˆH0 is the time-independent field-free Hamiltonian and
ˆ
V(t) = −ˆμ ⋅ E(t) is the time-dependent potential in the length gauge, written in terms of the molecular dipole and the time-dependent electric field E(t).18
We have considered a linearly polarized electric field E(t) along the α axis (α = x, y or z), representing a laser pulse,
E(t) = E0nαsin(ω0t + ϕ)f (t), (2) whereE0is the maximum field strength, nα is a unit vector along
the α axis, ω0is the carrier frequency, ϕ is the phase, and f (t) is the envelope function chosen as
f(t) = {cos 2(π
2σ(σ − t)) if ∣t − σ∣ ≤ σ
0 else, (3)
where σ is the width of the field envelope.
To solve Eq. (1), the wave function |Ψ(t)⟩ is expanded in a discrete basis of the eigenstates of the field-free Hamiltonian
ˆ
H0 composed of the ground state (k = 0) and all the excited states (k> 0),
∣Ψ(t)⟩ = ∑ k≥0
ck(t)∣Ψk⟩, (4)
whereck(t) are time-dependent coefficients. Inserting Eq.(4)into Eq.(1), and projecting on the eigenstates⟨Ψl|, gives the evolution equation
i∂c(t)
∂t = (H0+ V(t))c(t), (5)
where c(t) is the column matrix of the coefficients ck(t), H0 is the diagonal matrix of elements, i.e., H0,lk = ⟨Ψl∣ ˆH0∣Ψk⟩ = Ekδlk (whereEk is the energy of the eigenstatek), and V(t) is the non-diagonal matrix of elements, i.e., Vlk(t) = ⟨Ψl∣ ˆV(t)∣Ψk⟩. The initial wave function att = ti= 0 is chosen to be the field-free ground state, i.e.,ck(ti) = δk0. To solve Eq.(5), time is discretized and the split-propagator approximation is used, which reads as
c(t + Δt) ≈ e−iV(t)Δte−iH0Δtc(t), (6)
where Δt is the time step of the propagation. Since the matrix H0is diagonal,e−iH0Δtis also a diagonal matrix of elementse−iEkΔtδ
lk. The exponential of the non-diagonal matrix V(t) is calculated as
e−iV(t)Δt= U†e−iVd(t)ΔtU, (7)
where U is the unitary matrix describing the change of basis between the original eigenstates of ˆH0and a basis in which ˆV(t) is diagonal, i.e., Vd(t) = UV(t)U†.18,21
Once the time-dependent wavefunction |Ψ(t)⟩ is known, the time-dependent dipole μ(t) is computed as
μ(t) = ∑
lk
c∗l(t)ck(t)⟨Ψl∣ˆμ∣Ψk⟩ (8) from which, by taking the Fourier transform, the HHG spectrum is obtained, P(ω) = ∣∑ lk ⟨Ψl∣ˆμ∣Ψk⟩ 1 tf− ti∫ tf ti W(t)c∗l(t)ck(t)e−iωtdt∣ 2 , (9)
wheretiandtfare the initial and final propagation times andW(t) is the Hann (or Hanning) window function.
In the present work, we considered in Eq. (4), as truncated basis set, the molecular excited states described at the CIS and TDDFT level of theory. The extension of this approach to the solu-tion of the TDSE in the presence of a strong field brings to the RT-TD-CIS and RT-TDDFT methods that will be described in the following.
A. RT-TD-CIS
RT-TD-CIS is the time-dependent extension of the CIS method.51 In RT-TD-CIS, the time-dependent wave function is written on the truncated eigenstate basis of the CIS wave functions,
∣Ψ(t)⟩ = ∑ k=0
ck(t)∣ΨCISk ⟩. (10)
∣ΨCIS
k ⟩ are constructed by applying the excitation operator ˆR = r0+∑iaraiˆa†aˆaion the Hartree–Fock (HF) description of the field-free system, i.e.,∣ΨCISk ⟩ = ˆRk∣ϕHF0 ⟩. Here and throughout this paper, we use the indicesi and j for occupied orbitals, a and b for vir-tual ones, andp for generic orbitals. The operators ˆa†pand ˆap cre-ate and annihilcre-ate an electron in the orbital |ϕp⟩, and the ampli-tudesr0andrai are determined by solving the secular equation AX = ωCISX, where ωCIS= ECIS− EHF0 is the diagonal matrix of the exci-tation energies and X is the matrix of the CIS amplitudes (r0,rai) . EHF0 is the HF ground-state energy. The amplituderiarefers to the Slater determinant associated with the promotion of a single orbital from the occupied orbital i to the virtual a, while r0 is the CIS amplitude of the HF configuration. The matrix elements of A are given by
B. RT-TDDFT
The time-dependent wave function can be formally expanded in the field-free linear-response TDDFT states,
∣Ψ(t)⟩ = ∑ k=0
ck(t)∣ΨTDDFTk ⟩. (12)
As for CIS, the∣ΨTDDFTk ⟩ can be constructed by applying the exci-tation operator ˆR on the KS ground-state Slater determinant of the field-free system, i.e.,∣ΨTDDFTk ⟩ = ˆRk∣ϕKS0 ⟩.
Within the Tamm–Dancoff approximation (TDA), the TDDFT amplitudes are determined by solving AX = ωTDDFT/TDAX, where in this case, ωTDDFT/TDA= ETDDFT/TDA− EKS0 is the diagonal matrix of the excitation energies.EKS0 is the KS ground-state energy.
In the adiabatic approximation, the matrix elements of A are given by
Aia,jb= (ϵKSa − ϵKSi )δijδab+⟨aj∣wee∣ib⟩ + ⟨aj∣fxDFT∣ib⟩ + ⟨aj∣fcDFT∣ib⟩, (13) where ϵKSi and ϵKSa are the KS energies of occupied and virtual orbitals,⟨aj|wee|ib⟩ is the two-electron integral associated with the direct Coulomb, and⟨aj∣DFTx ∣ib⟩ and ⟨aj∣fcDFT∣ib⟩ are, respectively, the DFT exchange and correlation integrals, which can be calculated in different density functional approximations. In this work, we used forfxcDFTthe PBE functional, and its extension in real time will be labeled RT-TD-PBE.
When the range-separation approach is used, the matrix ele-mentAia,jbin Eq.(13)becomes
Aia,jb= (ϵKSa − ϵKSi )δijδab+⟨aj∣wee∣ib⟩ +⟨aj∣fxlr,ω,HF∣ib⟩ +⟨aj∣fxsr,ω,DFT∣ib⟩ +⟨aj∣fcDFT∣ib⟩, (14) where⟨aj∣fxlr,ω,HF∣ib⟩ = −⟨aj∣wlr,ωee ∣bi⟩ is the HF long-range Coulomb exchange and ⟨aj∣fxsr,ω,DFT∣ib⟩ is the short-range DFT Coulomb exchange. In the integrals, we explicitly indicated the depen-dence on the parameter ω, which controls the range-separation of the Coulomb electron–electron interaction. In this work, we used the LC-ωPBE scheme where the kernel is fxcLC-ωPBE = fxlr,ω,HF +fxsr,ω,DFT+fcDFT. For ω = 0, the scheme reduces to the usual linear-response TDDFT/TDA as in Eq.(13)in PBE. For ω =∞, the scheme reduces to CIS plus PBE correlation (PBEc). The extension in real time will be labeled RT-TD-LC-ωPBE.
III. COMPUTATIONAL METHODS
In RT-TD-CIS and RT-TDDFT (RT-TD-PBE or RT-TD-LC-ωPBE), the time-dependent wavefunction is expanded using a finite number of electronic excited states and transition dipole moments from the corresponding frequency-domain methods: CIS and TDDFT (PBE or LC-ωPBE). In the case of the LC-ωPBE func-tional, we used the range-separation parameter ω = 0.3 and ω = 0.4, following Refs.49and60. Moreover, in order to decouple the effect of exchange and correlation, we also calculated the CIS plus PBE correlation (CIS+PBEc) by adding the⟨aj∣fcDFT∣ib⟩ term in Eq.(11),
PBE only with exchange (PBEx) and LC-ωPBE only with exchange (LC-ωPBEx), and by removing the⟨aj∣fcDFT∣ib⟩ term from Eq.(13) and Eq.(14).
These frequency-domain methods propagated in time are labeled RT-TD-CIS+PBEc, RT-TD-PBEx, and RT-TD-LC-ωPBEx (with ω = 0.3 or 0.4). The TDDFT calculations were done within TDA. The dipole matrix elements were taken from field-free calculations by means of the Q-Chem software package70 and employed in an homemade code, Light,18,20,21,47,71 that propagates the wavepacket under the influence of a time-dependent strong field.
For all the theoretical methods employed, we used the compu-tational strategy we developed in the recent years, which demon-strated to be successful to describe HHG for atomic21,54,72and small molecular species.19,47We combined Gaussian continuum functions (K) with the double-d heuristic lifetime model.19,21,54,72,73This model prevents unphysical reflections of the wave function in the laser-driven dynamics. The heuristic lifetime model was originally pro-posed by Klinkuschet al.,73 and it consists of adding to the ener-gies calculated from the field-free Hamiltonian an imaginary term Γk= ∑ia∣ri,ka∣
2√2ϵHF/KS
a /d (ϵHF/KSa > 0), which represents the inverse lifetime. This approach is developed within singly excited schemes.22 The parameterd is empirical, representing the characteristic escape length that the electron is allowed to travel during the lifetime 1/Γk. Recently, we extended this model to two parameters (double-d heuristic lifetimes).19,21,54,72These parameters are chosen on the basis of the three-step model.12,13d0is equal to the maximum elec-tron excursion after ionization while d1< d0. Moreover, d0affects all the continuum states below the cutoff energy 1.32Ip + 3.17Up (Ipis the ionization potential energy and Upis the ponderomotive energy), while d1affects all the continuum states above the cutoff energy. The parameter d1is especially chosen to remove high-lying above-threshold states.19,21,54,72
We used the following Gaussian basis sets with K functions: 6aug-cc-pVTZ+6K for H2, 6aug-cc-pVTZ+3K for N2, and 6aug-cc-pVDZ+6K for CO2. The basis sets are reported in thesupplementary
material.
All results correspond to electronic dynamics at fixed nuclear geometries. To enable uniform comparison of the electron dynam-ics, we performed all calculations at the experimental equilib-rium distance of 1.400 a0 for H2, 2.074 a0 for N2, and 2.209 a0 for CO2.
We computed HHG spectra for a cos2-shaped laser field [see Eq.(3)] with carrier frequency ω0= 0.057 hartree (1.55 eV, 800 nm) and intensity I = 1014W/cm2(d0= 16.478 a0and d1= 1.414 a0). In parentheses, we reported the values of the escape length parameters used for the heuristic lifetime model. The laser was linearly polarized both perpendicular and parallel to the molecular axis. The duration of the pulse was 20 optical cycles (oc) (σ = 20 oc), where 1 oc = 2π/ω0. The time step was 0.24 as (0.01 a.u.).
TABLE I. Cutoff for the HHG spectrum with I = 1014W/cm2. H2 N2 CO2 RT-TD-CIS 25.92 30.20 30.76 RT-TD-CIS+PBEc 31.11 31.11 31.73 RT-TD-LC-ωPBEx (ω = 0.4) 24.44 27.22 27.47 RT-TD-LC-ωPBE (ω = 0.4) 25.06 28.07 28.40 RT-TD-LC-ωPBEx (ω = 0.3) 23.76 26.38 26.64 RT-TD-LC-ωPBE (ω = 0.3) 24.37 27.22 27.54 RT-TD-PBEx 20.41 22.82 23.07 RT-TD-PBE 18.85 23.65 23.97
TABLE II. Ionization potential energy (Ip) in Hartree for the different theoretical
schemes. Experimental values are also reported. In parentheses, we also reported the number of bound states, i.e., below the Ip, and of the number of continuum states,
i.e., above the Ip, used in the time-dependent propagation.
H2 N2 CO2 ICISp 0.594 (76/126) 0.779 (543/458) 0.803 (788/712) ICIS+PBEcp 0.623 (78/124) 0.818 (543/460) 0.845 (797/704) ILC-ωPBExp ω=0.4 0.530 (76/126) 0.650 (512/489) 0.661 (797/704) ILC-ωPBEp ω=0.4 0.557 (76/126) 0.687 (513/488) 0.701 (748/752) ILC-ωPBExp ω=0.3 0.501 (76/126) 0.614 (507/494) 0.625 (740/761) ILC-ωPBEp ω=0.3 0.527 (76/126) 0.650 (507/494) 0.664 (742/758) IPBExp 0.356 (9/193) 0.460 (330/671) 0.471 (529/972) IPBEp 0.381 (0/202) 0.496 (329/672) 0.510 (543/957) IExpp 0.56774 0.67375 0.71176
the intensity of the harmonics start to decrease until the signal goes out completely. The cutoff indicates that the extent of the HHG spectrum depends on the nature of the spectroscopic tar-get, via Ip, and also on the intensity and wavelength of the laser, via Up. InTable II, we also reported the Ipvalues at different lev-els of theory, together with the corresponding experimental ones. In the case of H2, the Ip is minus the HOMO energy, for N2is minus the HOMO-2 energy, and for CO2is the HOMO-3 energy. The choice of the Ipfor N2and CO2was motivated by the inclu-sion of all possible ionization channels that have been demon-strated participating in the electron dynamics generating the HHG spectrum. The HHG spectra of H2, N2, and CO2 with the per-pendicular pulse polarization are reported in the supplementary material.
IV. RESULTS AND DISCUSSION
In the following, we will attempt to answer the following question: how different theoretical descriptions of the electronic exchange and correlation can affect the shape of the HHG spectra of the H2, N2, and CO2molecular systems?
A. H2
To understand the role of exchange and correlation in HHG for the H2molecule, we started to analyze the truncated basis composed
FIG. 1. H2excitation energies only with exchange: bound (top, states 0–100) and
continuum (bottom, states 100–170) energies for CIS, LC-ωPBEx with ω = 0.3 and ω = 0.4, and PBEx. The Ipis also reported. The panels are a zoomed-in view of
the energy region of interest for the calculated HHG spectra.
FIG. 2. H2: HHG spectra only with exchange for RT-TD-CIS, RT-TD-LC-ωPBEx
FIG. 3. H2 excitation energies with
exchange and correlation: bound and onset of continuum energies for CIS vs CIS+PBEc, LC-ωPBEx vs LC-ωPBE
withω = 0.3 and ω = 0.4, and PBEx vs
PBE. The Ipis also reported.
of the ground- and excited-state correlated wave functions from the field-free Hamiltonian, which then define the time-dependent wavepacket in the RT-TD-CIS and in RT-TDDFT frameworks. In
Fig. (1), we compared the bound and the continuum energy states of the truncated basis calculated by CIS, LC-ωPBEx (ω = 0.4 and ω = 0.3), and PBEx. This comparison takes into account only the exchange contribution.
CIS is considered as the method of reference as it contains HF exchange with correct asymptotic behavior of the Coulomb potential (−1/r). The comparison with the other methods was done by taking the HF ground state energy as the zero energy reference. InFig. (1), for each theoretical methods, we also plotted the Ip(seeTable II). InTable II, we also reported the number of bound and continuum states for each theoretical scheme.
FIG. 4. H2: HHG spectra for RT-TD-CIS
vs RT-TD-CIS+PBEc, RT-TD-LC-ωPBEx
vs RT-TD-LC-ωPBE with ω = 0.3 and ω = 0.4, and PBEx vs
RT-TD-PBE. The laser has I = 1014W/cm2,ω
Considering the top panel of Fig. (1), we observe that the CIS methods satisfactorily represent the Rydberg states because of the correct asymptotic behavior of the Coulomb potential. The Ip in CIS (Table II) also has the best agreement with the exper-imental value compared to the other theoretical schemes. Con-sidering PBEx, for which the Coulomb potential decays expo-nentially at large distances, we observe that it does not support the Rydberg series of bound states. In this case, the Ip has the worst agreement with the experimental value when compared to the other theoretical calculations. This behavior is also supported by comparing the number of bound states of PBEx in Table II
with the other theoretical methods: the number of computed PBEx bound states is considerably lower. This is consistent with the wrong asymptotic behavior of the Coulomb exchange potential of PBEx.
The LC-ωPBEx contains a percentage of the HF long-range exchange and of the short-range PBE exchange. In the limit of ω =∞, the LC-ωPBEx scheme reduces to CIS, while for ω = 0,
FIG. 5. N2excitation energies only with exchange: bound (top, states 0–600) and
continuum (bottom, states 600–1000) energies for CIS, LC-ωPBEx with ω = 0.3
andω = 0.4, and PBEx. The Ipis also reported. The panels are a zoomed-in view
of the energy region of interest for the calculated HHG spectra.
the LC-ωPBEx scheme reduces to the PBEx. Therefore, by construc-tion, the LC-ωPBEx with ω = 0.4 and ω = 0.3 connects the CIS and the PBEx schemes. Considering ω = 0.4, the excitation energies are closer to the CIS, while for ω = 0.3, the excitation energies are closer to the PBEx. The corresponding Ipvalues inTable IIreflect the same behavior.
In the bottom panel ofFig. (1), the continuum energy states are compared. The trend is similar to the one observed for the bound states (top panel). However, despite the energy shift between the dif-ferent methods, the continuum energy density is the same for all the theoretical schemes.
The bound and continuum energy states represented inFig. (1)
are also the energy space in which the electrons move during the propagation, which can affect the description of the HHG spec-trum. InFig. (2), we show how these different electronic-structure descriptions impact the HHG spectrum of H2. For each level of the-ory, we also plotted the molecular energy cutoff12,13,19 reported in
Table I.
RT-TD-CIS, with the 6aug-ccpVTZ+6K basis set, reproduces well the main features of a HHG spectrum: perturbative/plateau, cutoff, and background regions.19,47It is interesting to compare this spectrum with other RT-TD-CIS calculations, which used different basis sets. Luppi and Head-Gordon18calculated the HHG spectrum for H2using various Gaussian basis sets, the largest one being d-aug-cc-pVTZ. The general trend of the HHG spectrum with this basis set is correct. However, for the region just beyond the cutoff, the com-parison with the 6aug-cc-pVTZ+6K basis set shows some differences due to the lack of diffuse functions and optimized K functions for the continuum states.
The HHG spectrum in RT-TD-CIS of the H2molecule was also calculated by Whiteet al.47using the cc-pVTZ and the 6aug-cc-pVTZ with additional basis function centers (ghost atoms). The 6aug-cc-pVTZ and 6aug-cc-pVTZ+6K give the same spectrum up to the 25th harmonic. Next, the lack of optimized Gaussian functions for continuum makes appear in the 6aug-cc-pVTZ a number of extra (artificial) harmonics, which are due to diffuseness of the basis set.
FIG. 6. N2: HHG spectra only with exchange for RT-TD-CIS, RT-TD-LC-ωPBEx
FIG. 7. N2 excitation energies with
exchange and correlation: bound and onset of continuum energies for CIS vs CIS+PBEc, LC-ωPBEx vs LC-ωPBE
withω = 0.3 and ω = 0.4, and PBEx vs
PBE. The Ipis also reported.
For this reason, Whiteet al.47found a second plateau, which is much more pronounced than what we found with the 6aug-cc-pVTZ+6K basis set.
The other approach used by White et al.47 to correct their basis for the continuum states was to insert ghost atoms in different geometrical configurations around the H2molecule. This technique permits to make disappear some of the spurious harmon-ics at high energy. Using this strategy makes the second plateau
less evident than with 6aug-cc-pVTZ, which is more in agreement with what we found with the 6aug-cc-pVTZ+6K basis set. Com-parison between the effects from ghost atoms and K functions is also reported for the HHG spectrum of the hydrogen atom in Ref.20.
The differences between our HHG spectrum and the one calcu-lated by Whiteet al.47with ghost atoms can also be due to the dif-ferent approach to treat ionization during the propagations. White
FIG. 8. N2: HHG spectra for RT-TD-CIS
vs RT-TD-CIS+PBEc, RT-TD-LC-ωPBEx
vs RT-TD-LC-ωPBE with ω = 0.3 and ω = 0.4, and PBEx vs
RT-TD-PBE. The laser has I = 1014 W/cm2, ω = 0.057 hartree, and polarization
et al.47used a heuristic lifetime model with only one escape length, at variance with what was done by us in this work (Sec.III). This means that the two HHG spectra differently exclude some recombination processes, which proceed through continuum states, including some classes of long-lived resonances.
In Fig. (2), we also report the HHG spectra computed by means of other theoretical approaches. The RT-TD-LC-ωPBEx with ω = 0.4 is the closest approach to RT-TD-CIS considering the description of the exchange. In fact, it contains a large percentage of HF exchange and a small percentage of PBE exchange. The effect is an overall increase in the spectrum intensity. The second plateau almost disappears. The RT-TD-LC-ωPBEx with ω = 0.3 has a higher percentage of the PBEx with respect to ω = 0.4. In addition, in this case, the spectrum intensity is generally increased with respect to RT-TD-CIS. However, some harmonics reappear at higher energy. Considering the HHG in PBEx, we observe that the peak intensities in the plateau region are not forcefully higher than with RT-TD-LC-ωPBEx with ω = 0.3, but the peaks are certainly more noisy. The most important feature with PBEx is that no second plateau is described. Some harmonics are very badly reproduced or not even resolved.
To go further with our analysis, we include the role of the PBE correlation (PBEc). InFig. (3), we show the bound and the onset of the continuum energy states calculated by CIS+PBEc, LC-ωPBE (ω = 0.3 and ω = 0.4), and PBE. The effect is the same for all the theoretical schemes. We obtain a rigid shift of the excitation energies and also of the Ip(seeTable II). It is interesting to note that for the PBE, we found no bound states.
InFig. (4), we analyzed the HHG for H2by comparing the the-oretical schemes with only exchange to the same schemes where we also included the PBEc. In the case of CIS, adding PBEc has very little impact. PBEc slightly lowers the intensity of some harmonics. Instead, PBEc has larger impact on LC-ωPBEx (ω = 0.3 and ω = 0.4) and PBEx. The harmonics in the plateau lower their intensities, but the strongest effect is shown for those harmonics in the region after the cutoff. PBEc seems to lower the background and therefore to better resolve high energy harmonics. Moreover, RT-TD-LC-ωPBE and RT-TDPBE are closer to RT-TD-CIS when PBEc is included. This behavior can be attributed to a compensation effect between the short-range PBEx and PBEc.
B. N2
To understand the role of exchange and correlation in HHG for the N2molecule, we compared inFig. (5)the bound and the continuum energy states calculated by CIS, LC-ωPBEx (ω = 0.4 and ω = 0.3), and PBEx. These are the energies of the truncated basis used in the RT-TD-CIS and in RT-TDDFT calculations. InFig. (5), we also plotted the Ip(seeTable II) for the different levels of theory. We remind that in this case, Ipis calculated as minus the HOMO-2 energy. As in the case of the H2molecule, the CIS method is our reference because of the correct asymptotic behavior of the electron Coulomb potential.
In the top panel ofFig. (5), the CIS shows a large number of bound states below the Ip. The behavior of the other theoretical approaches follows the same trend observed for the H2molecule. The plateau structures below Ipindicate the presence of other ion-ization channels, related to electron ionion-ization from HOMO and
HOMO-1. In the bottom panel ofFig. (5), the continuum energy states are compared. The trend is similar to the one observed for the bound states (top panel) but less regular than the trend for the continuum of the H2molecule.
InFig. (6), we show the HHG spectra for the different the-oretical schemes only including the electron exchange. The gen-eral trend already observed in H2is reproduced also for N2. HHG peaks by RT-TD-PBEx are slightly more intense and also more noisy than those described by RT-TD-CIS. The RT-TD-LC-ωPBEx with ω = 0.3 behaves close to RT-TD-PBEx, while RT-TD-LC-ωPBEx with ω = 0.4 is similar to RT-TD-CIS.
Luppi and Head-Gordon18 calculated the HHG spectrum of N2 in RT-TD-CIS with different Gaussian basis sets. The largest one was the d-aug-cc-pVTZ. From the comparison with our results, obtained with the 6aug-cc-pVTZ+3K basis set, we observe a simi-lar behavior of the two chosen computational protocols. However, we confirm that as for the H2 molecule, the critical point is the description of the energy region around the cutoff. The cutoff region
FIG. 9. CO2excitation energies only with exchange: bound (top, states 0–1000)
and continuum (bottom, states 1000–1500) energies for CIS, LC-ωPBEx with ω = 0.3 and ω = 0.4, and PBEx. The Ipis also reported. The panels are a
is poorly described by the d-aug-cc-pVTZ basis sets, which lacks a (large) number of diffuse functions and optimized K functions for the continuum states.
To go further in our analysis, we include the role of the PBE correlation (PBEc). InFig. (7), we show the bound and the onset of the continuum energy states calculated by CIS+PBEc, LC-ωPBE (ω = 0.3 and ω = 0.4), and PBE. The effect is the same for all the theoretical schemes. We obtain a rigid shift of the excitation energies and of the Ip(seeTable II).
InFig. (8), we analyzed the HHG for N2by comparing the the-oretical schemes with only exchange to the same schemes where we also included the PBEc term. In the case of RT-TD-CIS, adding PBEc has very little impact. PBEc slightly lowers the intensity of some harmonics. Instead, PBEc has larger impact on RT-TD-LC-ωPBEx (ω = 0.3 and ω = 0.4) and RT-TD-PBEx. The harmonics in the plateau lower their intensities, but the strongest effect is shown for those harmonics in the region after the cutoff. PBEc seems to lower the background and therefore to better resolve high energy harmon-ics. As for H2, RT-TD-LC-ωPBE and RT-TD-PBE are characterized by a better agreement with RT-TD-CIS when the PBE correlation is included.
C. CO2
InFig. (9), we show the bound and the continuum energy states calculated by CIS, LC-ωPBEx (ω = 0.4 and ω = 0.3), and PBEx. These are the energies of the truncated basis used in the RT-TD-CIS and in RT-TDDFT schemes. InFig. (9), we also plotted the Ip(see
Table II) for the different levels of theory employed here. Again, the CIS description of the CO2electronic structure can be considered as a theoretical reference, thanks to the correct asymptotic behavior of the Coulomb potential.
FIG. 10. CO2: HHG spectra only with exchange for RT-TD-CIS, RT-TD-LC-ωPBEx
withω = 0.3 and ω = 0.4, and RT-TD-PBEx. The laser has I = 1014W/cm2, ω = 0.057 hartree, and polarization parallel to the molecular axis.
In the top panel ofFig. (9), the behavior of the bound exci-tation energies, computed at the various levels of theory, is sim-ilar to the trend in H2and N2. The PBEx largely underestimates the ionization threshold and therefore the continuum energy col-lapse, whereas increasing the HF character implies a better descrip-tion of the long-range Coulomb potential and therefore of the bound excitation energies, which are then reproduced correctly together with the onset of the continuum. Another important obser-vation is that in the case of CO2, we observe energy plateau struc-tures, which are due to the presence of other ionization channels,
FIG. 11. CO2 excitation energies with
exchange and correlation: bound and onset of continuum energies for CIS vs CIS+PBEc, LC-ωPBEx vs LC-ωPBE
withω = 0.3 and ω = 0.4, and PBEx vs
FIG. 12. CO2: HHG spectra for
RT-TD-CIS vs RT-TD-RT-TD-CIS+PBEc,
RT-TD-LC-ωPBEx vs RT-TD-LC-ωPBE with ω = 0.3
andω = 0.4, and TD-PBEx vs
RT-TD-PBE. The laser has I = 1014W/cm2, ω = 0.057 hartree, and polarization
par-allel to the molecular axis.
corresponding to the electron removal from HOMO, HOMO-1, and HOMO-2.
In the bottom panel ofFig. (9), the continuum energy states are compared for the different theoretical schemes. In addition, in this case, the trend is the same as in H2and N2, the LC-ωPBEx energies with ω = 0.3 and 0.4 lie within the CIS and PBEx ones.
The effect of the different level of theory for electronic exchange and correlation employed to describe the CO2 wavepacket, and therefore, the HHG spectra is shown inFig. (10). RT-TD-CIS and RT-TD-LC-ωPBEx with ω = 0.3 and 0.4 have very similar behavior for the plateau, cutoff, and background region, except for very little differences in the peak intensity concerning the HHG in the plateau region, as already observed for H2 and N2. Instead, the behavior of RT-TD-PBEx is different. The intensity of the HHG spectrum is lower than with the other theoretical methods and, in particular, for the cutoff and background region. Moreover, it seems that har-monics continue to be present also at high energy, where instead the other methods do not describe any HHG anymore.
The effect of PBE correlation on top of the different theoretical schemes with only exchange is shown inFig. (11)for the excitation energies. The behavior is consistent with the trend found for H2and N2.
Including the correlation in the electronic-structure descrip-tion does not produce any appreciable or systematic change in the HHG spectra, as shown inFig. (12). The overall effect seems to be smaller than in H2and N2molecules.
D. Discussion
The physical mechanism at the origin of the HHG nonlinear process is usually described using the 3SM in which an electron is ionized by the laser pulse and subsequently driven far in the con-tinuum by the laser field before finally recombining with the parent
ion with the consequent emission of radiation. The 3SM relies on the single active electron (SAE) approximation, which supposes that it is only the outermost electron that contributes to the dynamics, while the other electrons are modeled by an effective potential. This implies that the electron correlation is not described and that the 3SM together with the SAE can only be qualitative.
For systems such as He where the correlation is small, the SAE can be a good approximation.77 However, for atomic sys-tems such as Be and Ne where the correlation is more important, it was necessary to go beyond the SAE and to use more accu-rate theoretical approaches.27In fact, using correlated methods per-mitted to describe for Be and Ne a second plateau well extended beyond the first one and also to identify a resonance peak above the plateau.78These features are a clear manifestation of electron correlation.30
In molecules, the description of the physics beyond the HHG spectroscopy is more complex than in atomic systems. In fact, together with the many-electron dynamics, there is also the possi-bility for the electrons to recombine with multiple atomic centers.79 An indication of the role of electronic correlation was pointed out by finding a clear correspondence with multiple orbital contributions to some specific spectral features of HHG for N2, O2, CO2, F2, N2O, and CO molecules.24,66,67,80Moreover, also for molecules, the many-electron dynamics brought evidence of a possible extension of the cutoff.45
descriptions was found. This implies that it is important to go beyond the SAE, but the HHG is not strongly sensitive to the way the electron exchange and correlation are treated. In the context of RT-TDDFT on a numerical grid, Chu and Memoli45compared the behavior of the exchange–correlation potential LBαand of the
local spin-density approximation with self-interaction correction (LSDA-SIC) to calculate HHG for the H2molecule. The exchange– correlation potentials are both corrected for the long-range behavior of the Coulomb potential. They found that the two methods repro-duce very similar HHG, which is in agreement with what we have obtained.
A surprising observation by comparing the different methods is that RT-TD-PBE (and RT-TD-PBEx), which wrongly reproduces the asymptotic behavior of the Coulomb potential, gives HHG spec-tra rather similar to RT-TD-CIS and RT-TD-LC-ωPBEx ones. Some differences are present in the cutoff and background region, but they are very small. This would indicate that the wrong asymp-totic behavior does not play a fundamental role (at least in our computational protocol). However, we want to point out that this result, which is also related to the self-interaction error, is still rather controversial.
Sunet al.81studied the molecular ion H+2. Their work focused on the numerical analysis of the self-interaction error in RT-TDDFT for H+2which has just one electron. Through a comparison with the exact solution of the time-dependent Schrödinger (TDSE) for H+2, they showed that LDA and PBE are in agreement with the TDSE for the lowest part of the spectrum, but spurious harmonics appear at higher energy. They also studied the performance of the LB94 and of the Fermi–Amaldi scheme plus PBE (LFAsPBE), which have the correct asymptotic behavior of the Coulomb potential.81They found that LB94 and LFAsPBE are in better agreement with exact TDSE, i.e., the asymptotic behavior is important. However, they also found that LFAsPBE is better than LB94, implying that the fine details of the exchange and correlation potential also affect the HHG spectra.
Concerning the HHG of the N2molecule, Macket al.82 com-pared the LDA and LB94 exchange–correlation potential. Both the approaches can reproduce the main features of the HHG spectra. The LDA has slightly higher intensity, but it is still predictive and accurate. Chu and Chu83 compared the HHG spectra of N
2using the LSDA without self-interaction correction and LBαand found 2–
3 order of magnitude of difference in the spectra. Therefore, they pointed out the importance of incorporating the correct asymp-totic long-range potential in the TDDFT treatment of strong field processes.45
Wardlow and Dundas84studied HHG in benzene, comparing LDA incorporating the Perdew–Wang parameterization of the cor-relation functional with and without self-interaction correction. The agreement between the two methods is very good. They observed an increase in the plateau harmonics and around the cutoff region, but for the method with the self-interaction correction.
The importance and the role of the long-range Coulomb poten-tial and the electron exchange and correlation merit further theo-retical investigations. The methods could have different sensitivity to the laser intensity and to the molecular systems. We believe the most critical point for all these methods is the description of the continuum and how it couples to the electron dynamics during the time-dependent propagation.
V. CONCLUSIONS
In this work, we studied the role of electronic exchange and correlation in the HHG spectroscopy of H2, N2, and CO2molecules using differentab initio electronic structures methods: RT-TD-CIS and RT-TDDFT (PBE and LC-ωPBE) using truncated basis sets composed of correlated wave functions from the corresponding field-free electronic Hamiltonian.
We computed HHG spectra for a cos2-shaped laser field with carrier frequency ω0= 0.057 hartree (1.55 eV, 800 nm) and inten-sity I = 1014W/cm2. We combined Gaussian continuum functions (K) and a heuristic lifetime model with two parameters for modeling ionization.19,21,54,72
We separated the effect of exchange from the effect of corre-lation by comparing the methods: RT-TD-CIS, RT-TD-LC-ωPBEx (ω = 0.3 and ω = 0.4), and RT-TD-PBEx, which contain only elec-tronic exchange. This permitted, without any bias, to observe the effect of the long-range HF and of the short-range PBE on HHG. The correlation was then added in the form of PBE correlation.
All the methods give very similar HHG spectra, and they seem not to be particularly sensitive to the different description of exchange and correlation or to the correct asymptotic behavior of the Coulomb potential. Despite this general trend, some differences are found in the energy region connecting the cutoff and the back-ground. Methods such as RT-TD-CIS, RT-TD-LC-ωPBEx (ω = 0.4), and RT-TD-LC-ωPBE (ω = 0.4), which contain long-range HF or at least a percentage of it, seem to better resolve the harmonics.
The investigation of electronic correlation in molecular HHG is a complex problem as the molecular continuum is coupled with strong field. This subject deserves further investigations, as, for example, the inclusion of double excitations and the analysis of other range-separated, hybrid and double-hybrid functionals.
SUPPLEMENTARY MATERIAL
See thesupplementary materialfor the HHG spectra of H2, N2, and CO2for the different levels of theory, computed with a pulse polarization perpendicular to the molecular axis. Basis sets in the Q-Chem format are also reported.
ACKNOWLEDGMENTS
C.F.P. acknowledges the Extra Erasmus program of the Univer-sity of Trieste and Sorbonne Universités for financial support. The authors acknowledge the Computational Center of the University of Trieste.
DATA AVAILABILITY
The data that support the findings of this study are avail-able within the article and itssupplementary materialand from the corresponding author upon reasonable request.
REFERENCES
1G. Sansone,Nat. Photonics14, 131 (2020). 2A. J. Uzanet al.,Nat. Photonics14, 188 (2020). 3T. Gaumnitzet al.,Opt. Express25, 27506 (2017).
5
A. Marciniaket al.,Nat. Commun.10, 337 (2019).
6
P. Peng, C. Marceau, and D. M. Villeneuve,Nat. Rev. Phys.1, 144 (2019).
7
M. Ossianderet al.,Nat. Phys.13, 280 (2017).
8
R. Cireasaet al.,Nat. Phys.11, 654 (2015).
9
M. Nisoli, P. Decleva, F. Calegari, A. Palacios, and F. Martín,Chem. Rev.117, 10760 (2017).
10J. P. Marangos,J. Phys. B: At., Mol. Opt. Phys.
49, 132001 (2016).
11A. Dubrouilet al.,Nat. Commun.
5, 4637 (2014).
12P. B. Corkum,Phys. Rev. Lett.
71, 1994 (1993).
13M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum,Phys.
Rev. A49, 2117 (1994).
14
R. Santra and A. Gordon,Phys. Rev. Lett.96, 073906 (2006).
15
A. Gordon, F. X. Kärtner, N. Rohringer, and R. Santra,Phys. Rev. Lett.96, 223902 (2006).
16S. Beaulieuet al., Phys. Rev. Lett. 117, 203001 (2016). 17F. Catoireet al., Phys. Rev. Lett. 121, 143902 (2018). 18E. Luppi and M. Head-Gordon,Mol. Phys.
110, 909 (2012).
19M. Labeyeet al.,J. Chem. Theory Comput.
14, 5846 (2018).
20E. Coccia and E. Luppi,Theor. Chem. Acc.
135, 43 (2016).
21E. Cocciaet al.,Int. J. Quantum Chem.
116, 1120 (2016).
22E. Coccia, R. Assaraf, E. Luppi, and J. Toulouse,J. Chem. Phys.
147, 014106 (2017).
23
J. Caillatet al.,Phys. Rev. A71, 012712 (2005).
24
M. Ruberti, P. Decleva, and V. Averbukh,Phys. Chem. Chem. Phys.20, 8311 (2018).
25M. Nest, R. Padmanaban, and P. Saalfrank, J. Chem. Phys.
126, 214106 (2007).
26
T. Sato and K. L. Ishikawa,Phys. Rev. A91, 023417 (2015).
27
T. Satoet al.,Phys. Rev. A94, 023405 (2016).
28
M. C. H. Wong, J.-P. Brichta, M. Spanner, S. Patchkovskii, and V. R. Bhardwaj, Phys. Rev. A84, 051403(R) (2011).
29F. Bedurke, T. Klamroth, P. Krause, and P. Saalfrank,J. Chem. Phys.
150, 234114 (2019).
30
I. Tikhomirov, T. Sato, and K. L. Ishikawa, Phys. Rev. Lett.118, 203202 (2017).
31P. Krause, T. Klamroth, and P. Saalfrank,J. Chem. Phys.
123, 074105 (2005).
32C. Huber and T. Klamroth,Appl. Phys. A
81, 93 (2005).
33P. Krause, T. Klamroth, and P. Saalfrank,J. Chem. Phys.
127, 034107 (2007).
34N. Rohringer, A. Gordon, and R. Santra,Phys. Rev. A
74, 043420 (2006).
35T. Sato and K. L. Ishikawa,Phys. Rev. A
88, 023402 (2013).
36N. Tancogne-Dejeanet al.,J. Chem. Phys.
152, 124119 (2020).
37T. P. Rossi, M. Kuisma, M. J. Puska, R. M. Nieminen, and P. Erhart,J. Chem.
Theory Comput.13, 4779 (2017).
38
P. Wopperer, P. M. Dinh, P.-G. Reinhard, and E. Suraud,Phys. Rep.562, 1 (2015).
39C.-Z. Gao, P. M. Dinh, P.-G. Reinhard, and E. Suraud,Phys. Chem. Chem. Phys. 19, 19784 (2017).
40
M. Vincendon, L. Lacombe, P. M. Dinh, E. Suraud, and P. G. Reinhard,Comput. Mater. Sci.138, 426 (2017).
41J. Pilmè, E. Luppi, J. Berés, C. Hoée-Levin, and A. de la Lande,J. Mol. Modell. 20, 2368 (2014).
42
X. Wuet al.,J. Chem. Theory Comput.13, 3985 (2017).
43
X. Wu, A. Alvarez-Ibarra, D. R. Salahub, and A. de la Lande,Eur. J. Phys. D72, 206 (2018).
44A. Pariseet al.,J. Phys. Chem. Lett.9, 844 (2018). 45X. Chu and P. J. Memoli,Chem. Phys.391, 83 (2011).
46
F. Ding, W. Liang, C. T. Chapman, C. M. Isborn, and X. Li,J. Chem. Phys.135, 164101 (2011).
47A. F. White, C. J. Heide, P. Saalfrank, M. Head-Gordon, and E. Luppi,Mol.
Phys.114, 947 (2016).
48
J. A. Sonk, M. Caricato, and H. B. Schlegel,J. Phys. Chem. A115, 4678 (2011).
49
J. A. Sonk and H. B. Schlegel,J. Phys. Chem. A115, 11832 (2011).
50
M. E. Casida, “Time-dependent density functional response theory for molecules,” inRecent Advances in Density Functional Methods, edited by D. P. Chong (World Scientific, Singapore, 1995), Part I, p. 155.
51
A. Dreuw and M. Head-Gordon,Chem. Rev.105, 4009 (2005).
52
T. Sato, H. Pathak, Y. Orimo, and K. L. Ishikawa,J. Chem. Phys.148, 051101 (2018).
53H. Pathak, T. Sato, and K. L. Ishikawa,J. Chem. Phys.
152, 124115 (2020).
54E. Coccia and E. Luppi,Theor. Chem. Acc.
138, 96 (2019).
55E. Runge and E. K. U. Gross,Phys. Rev. Lett.
52, 997 (1984).
56E. K. U. Gross and W. Kohn,Phys. Rev. Lett.
55, 2850 (1985).
57P. Elliott, S. Goldson, C. Canahui, and N. T. Maitra,Chem. Phys. 391, 110 (2011).
58
J. P. Perdew,Chem. Phys. Lett.64, 127 (1979).
59
J. P. Perdew and A. Zunger,Phys. Rev. B23, 5048 (1981).
60
O. A. Vydrov and G. E. Scuseria,J. Chem. Phys.125, 234109 (2006).
61
O. A. Vydrov, J. Heyd, A. V. Krukau, and G. E. Scuseria,J. Chem. Phys.125, 074106 (2006).
62J.-D. Chai and M. Head-Gordon,J. Chem. Phys.
131, 174105 (2009).
63F. Zapata, E. Luppi, and J. Toulouse,J. Chem. Phys.
150, 234104 (2019).
64R. van Leeuwen and E. J. Baerends,Phys. Rev. A
49, 2421 (1994).
65N. Gauriot, V. Véniard, and E. Luppi,J. Chem. Phys.
151, 234111 (2019).
66M. Monfared, E. Irani, and R. Sadighi-Bonabi,J. Chem. Phys.
148, 234303 (2018).
67
X. Chu and G. C. Groenenboom,Phys. Rev. A93, 013422 (2016).
68
T. T. Gormanet al.,J. Chem. Phys.150, 184308 (2019).
69
K. Kaufmann, W. Baumeister, and M. Jungen,J. Phys. B: At. Mol. Opt. Phys.22, 2223 (1989).
70Y. Shaoet al.,Phys. Chem. Chem. Phys.
8, 3172 (2006).
71E. Luppi and M. Head-Gordon,J. Chem. Phys.
139, 164121 (2013).
72E. Coccia,Mol. Phys.
118, e1769871 (2020).
73S. Klinkusch, P. Saalfrank, and T. Klamroth,J. Chem. Phys.
131, 114304 (2009).
74
D. Shiner, J. M. Gilligan, B. M. Cook, and W. Lichten,Phys. Rev. A47, 4042 (1993).
75A. Hamnett, W. Stoll, and C. E. Brion,J. Electron. Spectrosc. Related Phenom. 8, 367 (1976).
76
K. Kimura, S. Katsumata, Y. Achiba, T. Yamazaki, and S. Iwata,Handbook of HeI Photoelectron Spectra (Japan Scientific Societies Press, Tokyo und Halstead Press, New York, 1981).
77
R. Reiff, T. Joyce, A. Jaro´n-Becker, and A. Becker,J. Phys. Commun.4, 065011 (2020).
78Y. Li, T. Sato, and K. L. Ishikawa,Phys. Rev. A
99, 043401 (2019).
79M. Lein, N. Hay, R. Velotta, J. P. Marangos, and P. L. Knight,Phys. Rev. A 66, 023805 (2002).
80
J. Heslar, D. A. Telnov, and S.-I. Chu,Phys. Rev. A83, 043414 (2011).
81H.-L. Sun, W.-T. Peng, and J.-D. Chai,RSC Adv.
6, 33318 (2016).
82M. R. Mack, D. Whitenack, and A. Wasserman,Chem. Phys. Lett.558, 15
(2013).
83X. Chu and S.-I. Chu,Phys. Rev. A
64, 063404 (2001).
